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Asymptotic inference for an ising lattice. II

Published online by Cambridge University Press:  01 July 2016

D. K. Pickard
D. K. Pickard*
Affiliation:
The Australian National University
*
*Now at Harvard University.
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Abstract

In Pickard (1976) limit theorems were obtained for the classical Ising model at non-critical points. These determined the asymptotic distribution of the sample nearest-neighbour correlation, thereby providing a basis for statistical inference by confidence intervals. In this paper, these limit theorems are extended to the statistically significant case of different vertical and horizontal interactions. Results at critical points are also obtained. Critical points clearly have the potential to seriously distort statistical inferences, especially in their immediate neighbourhoods. For our Ising model it turns out that such distortion is relatively minor. Surprisingly, in the two-parameter case the correlation between the sufficient statistics exhibits peculiar asymptotic behaviour resulting in a singular covariance matrix at critical points in the central limit theorem. Finally, at critical points, unusual norming constants are required for the central limit theorem, and our results are much more sensitive to the relative rate at which m, n tend to infinity.

Keywords

AUTO-LOGISTIC MODELISING MODELGIBBS ENSEMBLENEAREST-NEIGHBOUR SYSTEMSSTATISTICAL INFERENCE ON LATTICESCENTRAL LIMIT THEOREMNEAREST-NEIGHBOUR CORRELATIONCRITICAL POINTS
Type
Research Article
Information
Advances in Applied Probability , Volume 9 , Issue 3 , September 1977 , pp. 476 - 501
Copyright
Copyright © Applied Probability Trust 1977 

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